Date of Original Version
Theory and Applications of Categories 15, 5 (2005): 147-163.
Abstract or Table of Contents
This paper constructs models of intuitionistic set theory in suitable categories. First, a Basic Intuitionistic Set Theory (BIST) is stated, and the categorical semantics are given. Second, we give a notion of an ideal over a category, using which one can build a model of BIST in which a given topos occurs as the sets. And third, a sheaf model is given of a Basic Intuitionistic Class Theory conservatively extending BIST. The paper extends the results in  by introducing a new and perhaps more natural notion of ideal, and in the class theory of part three.